A 46​.0-m guy wire is attached to the top of a 34.6​-m antenna and to a point on the ground. How far is the point on the ground from the base of the​ antenna, and what angle does the guy wire make with the​ ground?


This problem involves a right triangle, where:

- The guy wire is the hypotenuse (46.0 m),
- The height of the antenna is one leg of the triangle (34.6 m),
- The distance from the base of the antenna to the point where the guy wire touches the ground is the other leg.

### Step 1: Calculate the distance from the base of the antenna to the point on the ground (using the Pythagorean theorem).

We can use the Pythagorean theorem to find this distance \( d \):

\[
c^2 = a^2 + b^2
\]
Where:
- \( c = 46.0 \, \text{m} \) (the length of the guy wire),
- \( a = 34.6 \, \text{m} \) (the height of the antenna),
- \( b = d \, \text{m} \) (the distance we need to find).

\[
46.0^2 = 34.6^2 + d^2
\]

### Step 2: Solving for \( d \).

\[
2116 = 1197.16 + d^2
\]
\[
d^2 = 2116 - 1197.16 = 918.84
\]
\[
d = \sqrt{918.84} \approx 30.3 \, \text{m}
\]

So, the distance from the base of the antenna to the point on the ground is approximately **30.3 meters**.

### Step 3: Calculate the angle the guy wire makes with the ground.

Now, we can use trigonometry to find the angle \( \theta \) that the guy wire makes with the ground. Using the sine function:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{34.6}{46.0}
\]
\[
\theta = \sin^{-1}\left(\frac{34.6}{46.0}\right)
\]
\[
\theta \approx \sin^{-1}(0.7522) \approx 48.8^\circ
\]

So, the angle that the guy wire makes with the ground is approximately **48.8 degrees**.

### Final Answer:
- The distance from the point on the ground to the base of the antenna is approximately **30.3 meters**.
- The angle the guy wire makes with the ground is approximately **48.8 degrees**.

Would you like further details on any of these steps or concepts?

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Here are five related questions to expand on this topic:
1. How would the distance from the base of the antenna change if the guy wire were 50 meters long?
2. What happens to the angle if the antenna height increases but the guy wire length remains the same?
3. How do you use the tangent function to find the angle between the guy wire and the ground?
4. What if the ground isn't flat? How would you approach the problem?
5. Can you determine the total tension in the guy wire if the weight of the antenna is also known?

**Tip:** Always check if you can apply the Pythagorean theorem when working with right triangles in physics or engineering problems like this one.

A 46.0-m guy wire is attached to the top of a 34.6-m antenna and to a point on the ground. How far is the point on the ground from the base of the antenna, and what angle does the guy wire make with the ground?

Solve the problem of a 46.0-meter guy wire attached to a 34.6-meter antenna. Learn how to calculate the distance from the base of the antenna to the ground point and the angle the wire makes with the ground using the Pythagorean theorem and trigonometric functions. Suitable for students in Grades 9-12.

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